Search

The indirect production mechanism of doubly charmed tetraquark $T_{cc}$ through three decay channels, Higgs$/Z^{0}\to \langle cc\rangle[n] \to T_{cc} +\bar{c}+\bar{c} $ and $W^+\to \langle cc\rangle[n] \to T_{cc} +\bar{c}+\bar{s} $, are analyzed at LHC and CEPC within the framework of NRQCD. The spin and color quantum number $[n]$ of the intermediate $\langle cc\rangle$ diquark state can be $[^1 S_0]_6$ and $[^3 S_1]_{\bar{3}}$. Then the intermediate $\langle cc\rangle$ diquark evolves into tetraquark components via the fragmentation process by trapping two light antiquarks from the vacuum. After all the doubly charmed tetraquark components are summed, including $T_{cc}^{\bar{u} \bar{u}}$, $T_{cc}^{\bar{u} \bar{d}}$, $T_{cc}^{\bar{d} \bar{d}}$, $T_{cc}^{\bar{u}\bar{s}}$, and $T_{cc}^{\bar{d}\bar{s}}$, the total decay widths, branching ratios, and produced events per year for the production of $T_{cc}$ can be predicted at LHC and CEPC, respectively. The differential distributions and three main sources of theoretical uncertainty are also presented and discussed. The results show that the produced events per year for $T_{cc}$ via $W^{+}$ decays is $2.67\times10^5$, nearly two orders of magnitude larger than that by Higgs decays ($1.70\times10^{3}$) and $Z^{0}$ decays ($7.15\times10^3$) at LHC. However at CEPC, the largest contribution for the production of $T_{cc}$ is through $Z^{0}$ decays, about $2.42\times10^6$. There are only $7.33\times10^{-1}$ and $3.01\times10^{2}$ $T_{cc}$ events produced per year at CEPC through Higgs and $W^+$ decay, respectively., Comment: 15 pages, 2 figures, 6 tables

Multiplicity fluctuations play a crucial role in relativistic heavy-ion collisions. In this work, we explore how the multiplicity fluctuations can be effectively suppressed in the measurement of particle correlations. In particular, through proper normalization, particle correlations can be evaluated in a manner irrelevant to multiplicity. When the multiplicity fluctuations are adequately extracted, Monte Carlo simulations show that the remaining correlations possess distinct features buried in the otherwise overwhelming fluctuations. Moreover, we argue that such a normalization scheme naturally agrees with the multi-particle correlator, which can be evaluated using the Q-vectors. The implications of the present study in the data analysis are also addressed., Comment: 15 pages, 3 figures

In the framework of nonrelativistic QCD, the excited doubly heavy baryons are thoroughly studied via the channel $e^{+} e^{-}\rightarrow \langle QQ^{\prime}\rangle[n] \rightarrow \Xi_{QQ^{\prime}} +\bar{Q^{\prime}} +\bar{Q}$, which takes place at the collision energy $Z$-pole. $Q^{(\prime)}$ represents $b$ or $c$ quark for the production of $\Xi_{cc}$, $\Xi_{bc}$, and $\Xi_{bb}$, respectively. All of the intermediate diquark states $\langle QQ'\rangle[n]$ in $P$-wave, $\langle cc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$, $\langle cc\rangle[^{3}P_{J}]_{\mathbf{6}}$, $\langle bc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{3}P_{J}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb \rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$, and $\langle bb\rangle[^{3}P_{J}]_{\mathbf{6}}$ with $J=0$, 1, or 2, are taken into account. The cross sections and differential distributions, including the transverse momentum, rapidity, angular, and invariant mass, are discussed for the excited baryons production. We find that the contributions of $\langle cc \rangle$, $\langle bc \rangle$, and $\langle bb \rangle$ in $P$-wave are found to be 3.97$\%$, 5.08$\%$, and 5.89$\%$, respectively, compared to $S$-wave. Supposing that all excited states can decay into the ground state 100\%, the total events $N_{\Xi_{cc}}=8.48 \times10^{4-6}$, $N_{\Xi_{bc}}=2.26\times10^{5-7}$, and $N_{\Xi_{bb}}=4.12 \times10^{3-5}$ would be produced at the Super-$Z$ Factory with a high luminosity up to ${\cal L} \simeq 10^{34-36}{\rm cm}^{-2} {\rm s}^{-1}$., Comment: 15 pages, 4 figures, 5 tables, version to be published in EPJC

We explore the quasiparticle model at finite chemical potential related to Ru-Keng Su's distinguished contributions to the topic. Besides, we discuss recent developments in the model, and in particular, one argues that the effective mass of the quasiparticle might attain a specific form as a function of momentum, in addition to its dependence on temperature and chemical potential. Unlike the approaches based on the properties of underlying symmetry or renormalization group, the momentum dependence emerges as a special solution to an integro-differential equation resulting from the underlying thermodynamic consistency. Moreover, this special solution to the problem is shown to be more general than previously explored in the literature. Instead of fitting to the lattice QCD data at vanishing chemical potential, in this work, we adopt a ``bottom-up'' approach by assuming some analytic ansatzes that are manifestly thermodynamically consistent. The remaining physical quantities are subsequently derived, and possible implications are also addressed., Comment: 12 pages, 5 figures

Through the interaction of Higgs with heavy quarks in standard model, we have systematically studied and predicted the production of excited doubly heavy baryons based on non-relativistic QCD theory. The decay widths, differential distributions, and major theoretical uncertainties of the excited doubly heavy baryons via the process $H \rightarrow \langle QQ'\rangle[n] \rightarrow \Xi_{QQ'}+ \bar {Q'} + \bar {Q}$ are discussed in detail. The spin and color quantum number of the intermediate $P$-wave diquark state $\langle QQ'\rangle[n]$ can be $\langle cc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$, $\langle cc\rangle[^{3}P_{J}]_{\mathbf{6}}$, $\langle bc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{3}P_{J}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb \rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$ and $\langle bb\rangle[^{3}P_{J}]_{\mathbf{6}}$, with $J=0, 1, 2$. The contributions from all summed $P$-wave states can be about $3.05\%$, $3.23\%$ and $2.19\%$ of the $S$-wave states for the production of $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$, accordingly. Therefore, there will be about 0.41$\times10^4$ events of $\Xi_{cc}$, 6.35$\times10^4$ events of $\Xi_{bc}$ and 0.28$\times10^4$ events of $\Xi_{bb}$ produced per year at the HL-LHC, and a smaller number of events would be produced at the CEPC or ILC but with a cleaner background to be measured by the experiments., Comment: 17 pages, 4 figures, 6 tables. Accepted by European Physical Journal C

Within the framework of NRQCD, we calculate the production of excited doubly heavy baryons $\Xi_{bQ}$ through the semi-inclusive production process $t\rightarrow \langle bQ\rangle[n]\rightarrow \Xi_{bQ}+ \bar {Q} + W^+ $, where $Q= b$ or $c$ quark. The intermediate diquark state $\langle bQ\rangle[n]$ is in the excited $P$-wave state, including $[^1P_1]$ and $[^3P_J]$ ($J=$0, 1 or 2) in both color antitriplet state $\mathbf{\mathbf{\overline 3}}$ and color sixtuplet state $\mathbf{6}$, that is, $\langle bc\rangle[^{1}P_{1}]_{\mathbf{\overline 3}/ \mathbf{6}}$, $\langle bc\rangle[^{3}P_{J}]_{\mathbf{\overline 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}P_{1}]_{\mathbf{\overline 3}}$, and $\langle bb\rangle[^{3}P_{J}]_{\mathbf{6}}$. We find that the contributions from the P-wave states are about one order lower than the S-wave contributions, and this conclusion is consistent with others. We also analyze the invariant mass and angle differential distributions, and the theoretical uncertainty from the mass parameters and the renormalization scale. Finally, we can expect that about $1.14 \times10^{3-5}$ events of excited $\Xi_{bc}$ and $2.47 \times10^{1-3}$ events of excited $\Xi_{bb}$ can be produced per year at the LHC or HL-LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$., Comment: 11 pages, 5 figures, 7 tables. Some typos have been corrected and rewrite some sentences. arXiv admin note: text overlap with arXiv:1810.03834

It has been shown that the principle of maximum conformality (PMC) provides a systematic way to solve conventional renormalization scheme and scale ambiguities. The scale-fixed predictions for physical observables using the PMC are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. In the paper, we derive new degeneracy relations based on the renormalization group equations that involve both the usual $\beta$-function and the quark mass anomalous dimension $\gamma_m$-function, respectively. These new degeneracy relations lead to an improved PMC scale-setting procedures, such that the correct magnitudes of the strong coupling constant and the $\overline{\rm MS}$-running quark mass can be fixed simultaneously. By using the improved PMC scale-setting procedures, the renormalization scale dependence of the $\overline{\rm MS}$-on-shell quark mass relation can be eliminated systematically. Consequently, the top-quark on-shell (or $\overline{\rm MS}$) mass can be determined without conventional renormalization scale ambiguity. Taking the top-quark $\overline{\rm MS}$ mass ${\overline m}_t({\overline m}_t)=162.5^{+2.1}_{-1.5}$ GeV as the input, we obtain $M_t\simeq 172.41^{+2.21}_{-1.57}$ GeV. Here the uncertainties are combined errors with those also from $\Delta \alpha_s(M_Z)$ and the approximate uncertainty stemming from the uncalculated five-loop terms predicted through the Pad\'{e} approximation approach., Comment: 15 pages, 2 figures, to be published in Chinese Physics C

The {\it Principle of Maximum Conformality} (PMC), which generalizes the conventional Gell-Mann-Low method for scale-setting in perturbative QED to non-Abelian QCD, provides a rigorous method for achieving unambiguous scheme-independent, fixed-order predictions for physical observables consistent with the principles of the renormalization group. In addition to the original multi-scale-setting approach (PMCm), two variations of the PMC have been proposed to deal with ambiguities associated with the uncalculated higher order terms in the pQCD series, i.e. the single-scale-setting approach (PMCs) and the procedures based on ``intrinsic conformality" (PMC$_\infty$). In this paper, we will give a detailed comparison of these PMC approaches by comparing their predictions for three important quantities $R_{e^+e^-}$, $R_{\tau}$, and $\Gamma(H \to b \bar{b})$ up to four-loop pQCD corrections. The PMCs approach determines an overall effective running coupling $\alpha_s(Q)$ by the recursive use of the renormalization group equation, whose argument $Q$ represents the actual momentum flow of the process. Our numerical results show that the PMCs method, which involves a somewhat simpler analysis, can serve as a reliable substitute for the full multi-scale PMCm method, and that it leads to more precise pQCD predictions with small residual scale dependence., Comment: 17 pages, 6 figures, to be published in Nuclear Physics B

As one of the possible signals for the whereabouts of the critical point on the QCD phase diagram, recently, the multiplicity fluctuations in heavy-ion collisions have aroused much attention. It is a crucial observable of the Beam Energy Scan program of the Relativistic Heavy Ion Collider. In this work, we investigate the centrality dependence of the multiplicity fluctuations regarding the recent measurements from STAR Collaboration. By employing a hydrodynamical approach, the present study is dedicated to the noncritical aspects of the phenomenon. To be specific, in addition to the thermal fluctuations, finite volume corrections, and resonance decay at the freeze-out surface, the model is focused on the properties of the hydrodynamic expansion of the system and the event-by-event initial fluctuations. It is understood that the real signal of the critical point can only be obtained after appropriately subtracting the background, the latter is investigated in the present work. Besides the experimental data, our results are also compared to those of the hadronic resonance gas, as well as transport models., Comment: 10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1910.00705

Multiplicity fluctuations are one of the most crucial observables in the Beam Energy Scan program of the Relativistic Heavy Ion Collider. It is understood that they can be utilized to probe the whereabouts of the critical point on the phase diagram of the QCD matter. However, a significant portion of these fluctuations is, apart from that related to the QCD phase transition, attributed to the other origins, which we refer to as "noncritical" ones. The present study is dedicated to the noncritical aspects of the multiplicity fluctuations in heavy-ion collisions. In particular, we focus on those of dynamical origin, such as the hydrodynamic expansion of the system and the event-by-event initial fluctuations, in addition to the usual thermal fluctuations, finite volume corrections, and resonance decay at the freeze-out surface. The obtained results are compared to those of the hadronic resonance gas model as well as to the experimental data., Comment: 20 pages, 3 figures, 1 table

We systematically analyzed the production of semi-inclusive doubly heavy baryons ($\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$) for the process $H^0 \rightarrow \Xi_{QQ'}+ \bar {Q'} + \bar {Q}$ through four main Higgs decay channels within the framework of non-relativistic QCD. The contributions from the intermediate diquark states, $\langle cc\rangle[^{1}S_{0}]_{\mathbf{6}}$, $\langle cc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, $\langle bc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{1}S_{0}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}S_{0}]_{\mathbf{6}}$ and $\langle bb\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, have been taken into consideration. The differential distributions and three main sources of the theoretical uncertainties have been discussed. At the High Luminosity Large Hadron Collider, there will be about 0.43$\times10^4$ events of $\Xi_{cc}$, 6.32$\times10^4$ events of $\Xi_{bc}$ and 0.28$\times10^4$ events of $\Xi_{bb}$ produced per year. There are fewer events produced at the Circular Electron Positron Collider and the International Linear Collider, about $0.26\times 10^{2}$ events of $\Xi_{cc}$, $3.83\times 10^{2}$ events of $\Xi_{bc}$ and $0.17\times 10^{2}$ events of $\Xi_{bb}$ in operation., Comment: 15 pages, 3 figures, 7 tables

In the paper, we present a detailed discussion on the semi-inclusive production of doubly heavy baryons ($\Xi_{bc}$ and $\Xi_{bb}$) through top-quark decay channel, $t\rightarrow \Xi_{bQ^{\prime}}+ \bar {Q^{\prime}} + W^+ $, within the framework of non-relativistic QCD (NRQCD). In our calculations, the contributions from the intermediate diquark states, $\langle bc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{1}S_{0}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}S_{0}]_{\mathbf{6}}$ and $\langle bb\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, have been taken into consideration. Main uncertainties from the heavy quark mass $(m_c,~m_b~\mbox{or}~m_t)$, the renormalization scale $\mu_r$, and the nonperturbative transition probability have been estimated. For a comparison, we also analyze the production of doubly heavy baryons under the approximate fragmentation function approach. Estimated at the LHC or a High Luminosity LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$, there will be about $2.25\times10^{4-6}$ events of $\Xi_{bc}$ and $9.49\times10^{2-4}$ events of $\Xi_{bb}$ produced in one operation year through top-quark decays., Comment: 17 pages, 5 figures, 5 tables

In the paper, we discuss the possibility of observation of heavy quarkoniums via the processes involving flavor changing neutral currents (FCNC). More explicitly, we systematically calculate the production of heavy charmonium and $(c\bar{b})$-quarkonium through the top quark semi-exclusive rare FCNC decays in the framework of the non-relativistic QCD (NRQCD) factorization theory. Our results show that the total decay widths $\Gamma_{t\to \eta_c} =1.20^{+1.04+1.14}_{-0.51-0.45}\times 10^{-16}$ GeV, $\Gamma_{t\to J/\psi} =1.37^{+1.03+1.30}_{-0.51-0.51}\times 10^{-16}$ GeV, $\Gamma_{t\to B_c}=2.06^{+0.17+0.91}_{-0.17-0.54}\times 10^{-18}$ GeV, and $\Gamma_{t\to B^*_c}=6.27^{+0.63+2.78}_{-0.62-1.64}\times 10^{-18}$ GeV, where the uncertainties are from variation of quark masses and renormalization scales. Even though the decay widths are small, it is important to make a systematic study on the production of charmonium and $(c\bar{b})$-quarkonium through the top-quark decays via FCNC in the Standard Model, which will provide useful guidance for future new physics research from the heavy quarkonium involved processes., Comment: 17 pages, 8 figures and 6 tables, to be published in European Physical Journal C. arXiv admin note: text overlap with arXiv:1304.1303 by other authors

A hybrid parameterization of a quasiparticle equation of state is proposed, with a critical point implemented phenomenologically. On the one hand, a quasiparticle model with finite chemical potential is employed for the quark-gluon plasma phase, calibrated to the lattice quantum chromodynamics data. On the other hand, the low-temperature region for the hadronic phase of the matter is described by the hadronic resonance gas model with excluded volume correction. A particular interpolation scheme is adopted so that the phase transition is a smooth crossover for small chemical potential. A phenomenological critical pointed is implemented beyond which the phase transition becomes that of the first order., Comment: 6 pages, 1 figure. arXiv admin note: text overlap with arXiv:1709.03586

In this work, we revisit the thermodynamical self-consistency of the quasiparticle model with the finite baryon chemical potential adjusted to lattice QCD calculations. Here, we investigate the possibility that the effective quasiparticle mass is also a function of its momentum, $k$, in addition to temperature $T$ and chemical potential $\mu$. It is found that the thermodynamic consistency can be expressed in terms of an integro-differential equation concerning $k$, $T$, and $\mu$. We further discuss two special solutions, both can be viewed as sufficient condition for the thermodynamical consistency, while expressed in terms of a particle differential equation. The first case is shown to be equivalent to those previously discussed by Peshier et al. The second one, obtained through an ad hoc assumption, is an intrinsically different solution where the particle mass is momentum dependent. These equations can be solved by using boundary condition determined by the lattice QCD data at vanishing baryon chemical potential. By numerical calculations, we show that both solutions can reasonably reproduce the recent lattice QCD results of the Wuppertal-Budapest and HotQCD Collaborations, and in particular, those concerning finite baryon density. Possible implications are discussed., Comment: 18 pages, 4 figures

We propose a hybrid parameterization of a quasiparticle equation of state, where a critical point is implemented phenomenologically. In this approach, a quasiparticle model with finite chemical potential is used to describe the quark-gluon plasma phase by fitting to the lattice quantum chromodynamics data at high temperature. On the other hand, the hadronic resonance gas model with excluded volume correction is employed for the hadronic phase. An interpolation scheme is implemented so that the phase transition is a smooth crossover when the chemical potential is smaller than a critical value, or otherwise approximately of first order according to Ehrenfest's classification. Also, the thermodynamic consistency is guaranteed for the equation of state related to both the quasiparticle model and the implementation of the critical point., Comment: 13 pages, 4 figures

The Principle of Maximum Conformality (PMC) eliminates QCD renormalization scale-setting uncertainties using fundamental renormalization group methods. The resulting scale-fixed pQCD predictions are independent of the choice of renormalization scheme and show rapid convergence. The coefficients of the scale-fixed couplings are identical to the corresponding conformal series with zero $\beta$-function. Two all-orders methods for systematically implementing the PMC-scale setting procedure for existing high order calculations are discussed in this article. One implementation is based on the PMC-BLM correspondence \mbox{(PMC-I)}; the other, more recent, method \mbox{(PMC-II)} uses the ${\cal R}_\delta$-scheme, a systematic generalization of the minimal subtraction renormalization scheme. Both approaches satisfy all of the principles of the renormalization group and lead to scale-fixed and scheme-independent predictions at each finite order. In this work, we show that PMC-I and PMC-II scale-setting methods are in practice equivalent to each other. We illustrate this equivalence for the four-loop calculations of the annihilation ratio $R_{e^+ e^-}$ and the Higgs partial width $\Gamma(H\to b\bar{b})$. Both methods lead to the same resummed (`conformal') series up to all orders. The small scale differences between the two approaches are reduced as additional renormalization group $\{\beta_i\}$-terms in the pQCD expansion are taken into account. We also show that {\it special degeneracy relations}, which underly the equivalence of the two PMC approaches and the resulting conformal features of the pQCD series, are in fact general properties of non-Abelian gauge theory., Comment: 7 pages, 1 figure

A key problem in making precise perturbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling unambiguously at each finite order. The elimination of the uncertainty in setting the renormalization scale in pQCD will greatly increase the precision of collider tests of the Standard Model and the sensitivity to new phenomena. Renormalization group invariance requires that predictions for observables must also be independent on the choice of the renormalization scheme. The well-known Brodsky-Lepage-Mackenzie (BLM) approach cannot be easily extended beyond next-to-next-to-leading order of pQCD. Several suggestions have been proposed to extend the BLM approach to all-orders. In this paper we discuss two distinct methods. One is based on the "Principle of Maximum Conformality" (PMC), which provides a systematic all-orders method to eliminate the scale- and scheme- ambiguities of pQCD. The PMC extends the BLM procedure to all orders using renormalization group methods; as an outcome, it significantly improves the pQCD convergence by eliminating renormalon divergences. An alternative method is the "sequential extended BLM" (seBLM) approach, which has been primarily designed to improve the convergence of pQCD series. The seBLM, as originally proposed, introduces auxiliary fields and follows the pattern of the $\beta_0$-expansion to fix the renormalization scale. However, the seBLM requires a recomputation of pQCD amplitudes including the auxiliary fields; due to the limited availability of calculations using these auxiliary fields, the seBLM has only been applied to a few processes at low-orders......, Comment: 16 pages, 2 figures

The complete next-to-next-to-next-to-leading order short-distance and bound-state QCD corrections to $\Upsilon(1S)$ leptonic decay rate $\Gamma(\Upsilon(1S)\to \ell^+\ell^-)$ has been finished by Beneke {\it et al.} \cite{Beneke:2014qea}. Based on those improvements, we present a renormalization group (RG) improved pQCD prediction for $\Gamma(\Upsilon(1S)\to \ell^+\ell^-)$ by applying the principle of maximum conformality (PMC). The PMC is based on RG-invariance and is designed to solve the pQCD renormalization scheme and scale ambiguities. After applying the PMC, all known-type of $\beta$-terms at all orders, which are controlled by the RG-equation, are resummed to determine optimal renormalization scale for its strong running coupling at each order. We then achieve a more convergent pQCD series, a scheme- independent and more accurate pQCD prediction for $\Upsilon(1S)$ leptonic decay, i.e. $\Gamma_{\Upsilon(1S) \to e^+ e^-}|_{\rm PMC} = 1.270^{+0.137}_{-0.187}$ keV, where the uncertainty is the squared average of the mentioned pQCD errors. This RG-improved pQCD prediction agrees with the experimental measurement within errors., Comment: 11 pages, 4 figures. Numerical results and discussions improved, references updated, to be published in JHEP

As one of the key components of perturbative QCD theory, it is helpful to find a systematic and reliable way to set the renormalization scale for a high-energy process. The conventional treatment is to take a typical momentum as the renormalization scale, which assigns an arbitrary range and an arbitrary systematic error to pQCD predictions, leading to the well-known renormalization scheme and scale ambiguities. As a practical solution for such scale setting problem, the "Principle of Minimum Sensitivity" (PMS), has been proposed in the literature. The PMS suggests to determine an optimal scale for the pQCD approximant of an observable by requiring its slope over the scheme and scale changes to vanish. In the paper, we present a detailed discussion on general properties of PMS by utilizing three quantities $R_{e^+e^-}$, $R_\tau$ and $\Gamma(H\rightarrow b\bar{b})$ up to four-loop QCD corrections. After applying the PMS, the accuracy of pQCD prediction, the pQCD convergence, the pQCD predictive power and etc., have been discussed. Furthermore, we compare PMS with another fundamental scale setting approach, i.e. the Principle of Maximum Conformality (PMC)... Our results show that PMS does provide a practical way to set the effective scale for high-energy process, and the PMS prediction agrees with the PMC one by including enough high-order QCD corrections, both of which shall be more accurate than the prediction under the conventional scale setting. However, the PMS pQCD convergence is an accidental, which usually fails to achieve a correct prediction of unknown high-order contributions with next-to-leading order QCD correction only, i.e. it is always far from the "true" values predicted by including more high-order contributions., Comment: 15 pages, 9 figures. Eq.(9) corrected